Tendency of a Curve

Tendency of a curve is a discrete parameter, whose values are chosen amongst {+1,0,-1}, and is determined according to the tendency indicator vector of the curve, , where:

}, }, h0±

are the one-sided detachments of the functions x,y that form the curve, if these exist.

In this demonstration, all possible cases for the tendency indicator vector are depicted, and the tendency is shown once the vector is selected. The user may choose to set the tendecny indicator vector either in a discrete manner ("T.I.V" - tendency indicator vector), or in a continuous manner ("graphically"). The calculation of each of the parameters refers to the highlighted point (the bold pink one in the middle of the figure). The classification of the point according to the tendency indicator vector (slanted corner, perependicular edge etc.) is also updated.

Note that in case the "show tendency" checkbox is selected, then along with the notation of the tendency in the upper part of the graph, dashed corners appear on the left hand side of the curve at the point . The sum of the numbers in these corners is the curve's tendency; thus, the geometric interpretation of the tendency - i.e., the sum of the numbers in the dashed corners - is also depicted, to allow an intuitive understanding of this parameter's definition.

Thumbnail: Tendency at perpendicular corners consolidates with the definiton of the parameter αD from the discrete Green's theorem.

Snapshot 1: Tendency at acute corners whose angle is less than can be zero: in this case, the geometric interpration shows why. No dashed corners can be contained in the left hand-side of the curve at the pink point, hence the sum of the matching numbers is vacuouslyzero, and so is the tendency at the point.

Snapshot 2: Tendency is defined only in cases where the curve is defined for left and right neighborhoods of the given point.

Snapshot 3: Tendency at slanted corners is independent of the orientation of the curve, hence the bidirectional arrow. In this case we see that the sum of the numbers in the dashed corners is 1+(-1)+1=1, and so is the tendency. Note that here the user chose the endpoints in a graphical modes, which enabled the shorter edge (although the edges' length is not a critical parameter in this demonstration).

The discrete Green's theorem points out the connection between the double integral of a function over a domain which is a unification of rectangles, and a linear combination of the values of the given function's comulative distribution function, at the corners of the domain. The coefficients in this linear combination are uniquely determined according to the corner type - which is denoted by αD in the formulation of the theorem. The definition of tendency consolidates with the definition of αD, gives a rigorous interpretation to it via the tendency indicator vector at a point, and extends the definition of αD - to any kind of point (not only perpendicular corners as in the formulation of the discrete Green's theorem). The definition of tendency hence enables the discussion regarding an extended version of the discrete Green's theorem, which holds for more general types of domains.